February 16, 2018 | David F. Coppedge

Basener and Sanford Defend Paper Critiquing Fisher’s Theorem

Bill Basener and John Sanford recently critiqued Fisher’s Theorem, a long-trusted model for neo-Darwinism. Two evolutionary geneticists responded. Here is Part I their counter-response.

Defending the validity and significance of the new theorem “Fundamental Theorem of Natural Selection With Mutations, Part I: Fisher’s Impact

– Bill Basener and John Sanford

Joe Felsenstein and Michael Lynch (JF and ML) wrote a blog post, “Does Basener and Sanford’s model of mutation vs selection show that deleterious mutations are unstoppable?”  Their post is thoughtful and we are glad to continue the dialogue. This is the first part of a response to their post, focusing on the impact of R. A. Fisher’s work.  Our paper can be found at: https://link.springer.com/article/10.1007/s00285-017-1190-x

First, a short background on our paper:

The primary thesis of our paper is that Fisher was wrong, in a fundamental way, in his belief that his theorem (“The Fundamental Theorem of Natural Selection”), implied the certainty of ongoing fitness increase. His claim was that mutations continually provide variance, and selection turns the variance into fitness increase.  Central to his logic was that collectively; mutations have a net zero effect on fitness.  While Fisher assumed mutations are collectively fitness-neutral, it is now known that the vast majority of mutations are deleterious. So mutations can potentially push fitness down – even in the presence of selection.

Additionally, we provided a new mathematical model for the process of mutation and selection over time, which comes in an infinite population version and a finite population version.  The infinite population version uses a classical differential equations mutation/selection framework, with multiple reproducing subpopulations and mutations occurring between subpopulations, but incorporating a probabilistic distribution for mutation effects.  The finite version is obtained by adding the constraint that any subpopulation with less than one organism is assumed to have no members.

Our model is backed by a literature review in Section 2 of our paper (covering 9 pages with 71 citations), with Section 2.2 discussing previous infinite populations models and Section 2.3 focusing on finite ones.  Our  model is new in that it includes an arbitrary distribution of mutational effects, and we do not assume mutations are 50/50 beneficial /deleterious (as did Fisher), and we did not assume that all mutations have the same fixed effect (as with Lynch’s finite population models).

Part I: Ronald A. Fisher’s Impact – First, let’s discuss Fisher and the critique by Felsenstein and Lynch of our work on Fisher’s Theorem, and its historical importance:

In their critique, JF and ML do not dispute our logic regarding Fisher’s Theorem, but provide their perspective that Fisher’s contribution to population genetics and evolutionary theory was very limited. They say,

“One of us (JF) has argued at The Skeptical Zone that they have misread the literature on population genetics. The theory of mutation and natural selection developed during the 1920s, was relatively fully developed before Fisher’s 1930 book. Fisher’s FTNS has been difficult to understand, and subsequent work has not depended on it. But that still leaves us with the issue of whether the B and S simulations show some startling behavior…”

We respectfully disagree with their perspective that Fisher’s book and Theorem had only a minor impact. The book they refer to is “Genetical Theory of Natural Selection” (GTNS), which is where he published his FTNS Theorem.  To begin with, Google Scholar lists that Fisher’s book, GTNS, has been cited 20,254 times – this is not insignificant.

Below are some quotes from standard sources and leaders in the field that are consistent with our view that Fisher’s work, that his book GTNS contributed significantly to establishing the field of Population Genetics and his theorem FTNS was central to establishing Neo-Darwinian Theory.

From the book Philosophy of Biology, (Section on Fisher written by Robert A. Skipper, Jr., 2007, p.44):

“The Genetical Theory of Natural Selection is a point of departure in contemporaneous evolutionary thought, responsible in part for the origination of theoretical population genetics and what is commonly called the “modern synthetic theory of evolution.”

this continues,

“To be sure, Fisher’s work in statistics was revolutionary at the field’s conceptual foundation.  Moreover, Fisher’s work in genetics, highlighted mainly by his 1930 The Genetical Theory of Natural Selection, would, with good company in Haldane and Wright, revolutionize biology.”

In his textbook Population Genetics, M. Hamilton writes:

“Fisher’s 1930 book The Genetical Theory of Natural Selection established a rigorous mathematical framework that coupled Mendelian inheritance and Darwin’s quantitative model of natural selection.” (p. 204)

And then calls Fisher’s book,

“the first comprehensive treatment of natural selection that came out of the modern synthesis” (p. 206)

In the book The Logic of Chance: The Nature and Origin of Biological Evolution, author Eugene V. Koonin (who has authored over 600 articles, is Senior Investigator at NIH, and editor-in-Chief of the journal Biology Direct) writes:

“The foundations for the critically important synthesis of Darwinism and genetics were set in the late 1920s and early 1930s by the trio of outstanding theoretical geneticists: Ronald Fisher, Sewall Wright, and J. B. S. Haldane. They applied rigorous mathematics and statistics to develop an idealized description of the evolution of biological populations. The great statistician Fisher apparently was the first to see that, far from damning Darwinism, genetics provided a natural, solid foundation for Darwinian evolution. Fisher summarized his conclusions in the seminal 1930 book The Genetical Theory of Natural Selection (Fisher, 1930), a tome second perhaps only to Darwin’s Origin in its importance for evolutionary biology.5 This was the beginning of a spectacular revival of Darwinism that later became known as Modern Synthesis (a term mostly used in the United States) or neo-Darwinism (in the British and European traditions).”

In the book The Mathematics of Darwin’s Legacy, P. Schuster writes,

“Ronald Fisher, the great scholar of population genetics, presented the first mathematical unification of Darwin’s theory of natural selection and Mendel’s laws of inheritance [25].”

In the abstract to the paper Fisher’s fundamental theorem of natural selection in Trends in Ecology and Evolution (Frank and Slatkin 1992):

“Fisher’s Fundamental Theorem of natural selection is one of the most widely cited theories in evolutionary biology.”

In his textbook, Theoretical Evolutionary Genetics, Joe Felsenstein writes,

“Population genetics theory had its major developments in the 1920s-1940s (at the hands of Fisher, Wright, and Haldane)” (p. xvii)

The Wikipedia article on Fisher describes his contribution:

“In genetics, his work used mathematics to combine Mendelian genetics and natural selection; this contributed to the revival of Darwinism in the early 20th century revision of the theory of evolution known as the modern synthesis.”

In 1930, The Genetical Theory of Natural Selection was first published by Clarendon Press and is dedicated to Leonard Darwin. A core work of the neo-Darwinian modern evolutionary synthesis,[29] it helped define population genetics, which Fisher founded alongside Sewall Wright and J. B. S. Haldane”.

The paper What was Fisher’s fundamental theorem of natural selection and what was it for? In Studies in History and Philosophy of Biological and Biomedical Sciences (Plutynski, 2006) is probably the best thoroughly researched single source on Fisher’s theorem says:

“Fisher (1918, 1922) proposed a new way of picturing populations of organisms…. Starting with this novel conception, Fisher, Haldane, and Wright developed models of the genetics of populations.”

And then,

“Moreover, this analogy allowed Fisher to vindicate Darwin’s theory of natural selection, not by empirical demonstration, but by a mathematical argument to the effect that evolution was not only possible, but also necessary” p.75.

The quotes above from Plutynski give some good perspective on Fisher’s different contributions.  His 1918,1922 papers were foundational to population genetics.  His 1930 GTNS book was an additional significant contribution to population genetics, and his FTNS was seen as vindicating Darwin’s theory, giving what was perceived as a rigorous argument that (Darwinian) evolution is necessary, given Mendelian genetics.  The Plutynski paper is an insightful read, describing Fisher’s goals as follows:

“Fisher’s book, and the theorem in particular, is best understood as a continuation of his attempt to breach the divide between biometricians and Mendelians concerning the nature of heredity and the effectiveness of Darwinian selection. His motivation in almost all his work was to explain how it was possible to resolve this conflict, and to vindicate Darwinian selection as both a plausible and necessary cause of evolutionary change.”

Alan Grafen provides the following quote from William Hamilton describing Fisher’s Genetical Theory of Natural Selection:

“This is a book which, as a student, I weighed as of equal importance to the entire rest of my undergraduate BA course and, through the time I spent on it, I think it notched down my degree. Most chapters took me weeks, some months; … And little modified by molecular genetics, Fisher’s logic and ideas still underpin most of the ever broadening paths by which Darwinism continues its invasion of human thought. For a book that I rate only second in importance in evolution theory to Darwin’s ‘‘Origin’’ (this as joined with its supplement, ‘‘of Man’’), and also rate as undoubtedly one of the greatest books of the present century, the appearance of a variorum version is a major event. … By the time of my ultimate graduation, will I have understood all that is true in this book and will I get a First? I doubt it. In some ways some of us have overtaken Fisher; in many, however, this brilliant, daring man is still far in front.’” (Hamilton; dust-jacket of Fisher, 1930b).

Richard Dawkins ranks Fisher the greatest biologist since Darwin:

“Who is the greatest biologist since Darwin? That’s far less obvious, and no doubt many good candidates will be put forward. My own nominee would be Ronald Fisher. Not only was he the most original and constructive of the architects of the neo-Darwinian synthesis. Fisher also was the father of modern statistics and experimental design. He therefore could be said to have provided researchers in biology and medicine with their most important research tools, as well as with the modern version of biology’s central theorem.”

These quotes from a variety of sources support the tenet stated in the first sentences of our paper:

“R. A. Fisher was one of the greatest scientists of the 20th century. He is considered to be the singular founder of modern statistics and simultaneously the principle founder of population genetics (followed by Haldane and Wright). Fisher was the first to establish the conceptual link between natural selection and Mendelian genetics. This paved the way for what is now called neo-Darwinian theory.” – Basener and Sanford, 2017

Central to our paper is that Fisher’s theorem, which Dawkins calls “biology’s central theorem” does not imply what Fisher thought it did (and by extension what many others thought it did).  To clarify Fisher’s error, we distinguish between Fisher’s actual theorem (what he actually proved), and “Fisher’s Corollary”, which was unproven, and was really just an informal thought experiment based on his assumption that mutations have zero net effect on fitness. This corollary has clearly been falsified, which is essential to the popular concept that mutations simply supply genetic variance and natural selection converts this variance into ongoing increased fitness.

While Fisher’s Theorem is mathematically correct, his Corollary is false.  The simple logical fallacy is that Fisher stated that mutations could effectively be treated as not impacting fitness, while it is now known that the vast majority of mutations are deleterious, providing a downward pressure on fitness.  Our model and our correction of Fisher’s theorem (The Fundamental Theorem of Natural Selection with Mutations), take into account the tension between the upward force of selection with the downward force of mutations.

Our correction challenges a tradition central tenet of Neo-Darwinism.  It is often viewed that the upward force of selection acts without consideration of the downward force of mutations.  See, for example, the quote from Gould still taken as :

“The core of this synthetic theory restates the two most characteristic assertions of Darwin himself: first, that evolution is a two-stage process (random variation as raw material, natural selection as a directing force); secondly, that evolutionary change is generally slow, steady, gradual, and continuous. . . Orthodox neo-Darwinians extrapolate these even and continuous changes to the most profound structural transitions in life.” (Gould 1980)

We agree (apparently) with JF and ML that Fisher’s theorem has serious flaws.  The theorem was not clearly written, and is mathematically correct but with very limited application.  One of the most respected papers on Fisher’s FTNS was written by G. Price in 1972, which we quote in our paper as follows:

“Also, he [Fisher] spoke of the “rigour” of his derivation of the theorem and of “the ease of its interpretation”. But others have variously described his derivation as “recondite” (Crow and Kimura 1970), “very difficult” (Turner 1970), or “entirely obscure” (Kempthorne 1957). And no one has ever found any other way to derive the result that Fisher seems to state. Hence, many authors (not reviewed here) have maintained that the theorem holds only under very special conditions, while only a few (e.g.. Edwards 1967) have thought that Fisher may have been correct—if only we could understand what he meant! …here that this latter view is correct. Fisher’s theorem does indeed hold with the generality that he claimed for it. The mystery and the controversy result from incomprehensibility rather than error.”

At issue is that even though the flaws in Fisher’s theorem have been knowable for a long time, it has still been used as strong support for Neo-Darwinian Theory.  In the quote from Gould above, mutations add raw material of variation, and selection turns variation into increase in fitness (evolutionary progress); selection is assumed to act as an upward force without considering the downward force of mutations.  This is strongly linked to Fisher’s Corollary.  Despite the known flaws with Fisher’s work (known at least within the population genetics community), there is still a common perception that selection acts on mutations to maximize fitness as Fisher described.  As stated in the abstract of a recent 2016 paper Natural Selection and the maximization of fitness, on expansions of Fisher’s FTNS and Darwinism, by J Birch of the University of Cambridge:

“The notion that natural selection is a process of fitness maximization gets a bad press in population genetics, yet in other areas of biology the view that organisms behave as if attempting to maximize their fitness remains widespread. “


R. A. Fisher was one of the three founders of population genetics, and is considered by many to be the first and primary founder. His Fundamental Theorem of Natural Selection contributed significantly to a “revival of Darwinism” (see Koonin quote above and Wikipedia).  His theorem has been considered by many a significant and rigorous support for the Neo-Darwinian Theory (see quotes above).

Our paper shows that Fisher’s corollary is clearly false, and that he misunderstood the implications of his own theorem. He incorrectly believed that his theorem was a mathematical proof that showed that natural selection plus mutation will necessarily and always increase fitness. He also believed his theorem was on a par with a natural law (such as entropic dissipation and the second law of thermodynamics).  Because Fisher did not understand the actual fitness distribution of new mutations, his belief in the application of his “fundamental theorem of natural selection” was fundamentally and profoundly wrong – having little correspondence to biological reality. Therefore, we have reformulated Fisher’s model and have corrected his errors, thereby have established a new theorem that better describes biological reality, and allows for the specification of those key variables that will determine whether fitness will increase or decrease.

To be continued.

Note: This response has also been posted on The Skeptical Zone, where Felsenstein and Lynch initially posted their critique.

Update 3/20/18: Joe Felsenstein has issued a response to two of three points at The Skeptical Zone.

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  • Bill Neish says:

    Wow. Isn’t this something critics have been pointing out for decades that mutations are not neutral and only now someone is coming to grips with it in the Darwine camp? Amazing… Actually it is interesting to finally be seeing more and more scientists speaking out and even abandoning Darwine for ID. I think that is emboldens even more to reconsider their assumptions. Kind of exciting..

    Keep up the service… David. Hope to meet you one day.

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